数组的边界元素可能的最长递增序列

给定一个由正整数组成的长度为N的数组arr[] ，任务是找到可以由数组两端的元素组成的最长递增子序列。

例子 ：

Input: N=4 arr[] ={ 1, 4, 2, 3 }
Output: 1 3 4
Explanation: 
Append arr[0] to the sequence. Sequence = {1}. Array = {4, 2, 3}.
Append arr[2] to the sequence. Sequence = {1, 3}. Array = {4, 2}.
Append arr[0] to the sequence. Sequence = {1, 3, 4}. Array = {2}.
Therefore, {1, 3, 4} is the longest increasing sequence possible from the given array.

Input: N=3 arr[] ={ 4, 1, 3 }
Output: 3 4

方法：解决这个问题的想法是使用两个指针的方法。为严格递增序列中最右边的元素维护一个变量，比如rightmost_element 。在数组的末尾保留两个指针，分别说i和j并执行以下步骤，直到i超过j或两端的元素都小于rightmost_element ：

• 如果 arr[i] > arr[j]：
  • 如果 arr[j] > rightmost_element：设置rightmost_element = arr[j]并递减j 。将arr[j]添加到序列中。
  • 如果 arr[i] > rightmost_element：设置rightmost_element = arr[i]并增加i 。将arr[i]添加到序列中。
• 如果 arr[i] < arr[j]：
  • 如果 arr[i] > rightmost_element ：设置rightmost_element = arr[i]并增加i 。将arr[i]添加到序列中。
  • 如果 arr[j] > rightmost_element ：设置rightmost_element = arr[j]并递减j 。将arr[j]添加到序列中。
• 如果 arr[j] = arr[i]：
  • 如果 i = j：
    • 如果 arr[i]>rightmost_element ：将arr[i]添加到序列中。
  • 否则：分别勾选两端可以添加的最大元素，分别设为max_left和max_right 。
    • 如果 max_left > max_right ：从左端添加所有可以添加的元素。
    • 否则：添加所有可以从右侧添加的元素。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
#include 
using namespace std;
 
// Function to find longest strictly
// increasing sequence using boundary elements
void findMaxLengthSequence(int N, int arr[4])
{
    // Maintains rightmost element
    // in the sequence
    int rightmost_element = -1;
 
    // Pointer to start of array
    int i = 0;
 
    // Pointer to end of array
    int j = N - 1;
 
    // Stores the required sequence
    vector sequence;
 
    // Traverse the array
    while (i <= j) {
 
        // If arr[i]>arr[j]
        if (arr[i] > arr[j]) {
 
            // If arr[j] is greater than
            // rightmost element of the sequence
            if (arr[j] > rightmost_element) {
 
                // Push arr[j] into the sequence
                sequence.push_back(arr[j]);
 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else if (arr[i] > rightmost_element) {
 
                // Push arr[i] into the sequence
                sequence.push_back(arr[i]);
 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
            else
                break;
        }
 
        // If arr[i] < arr[j]
        else if (arr[i] < arr[j]) {
 
            // If arr[i] > rightmost element
            if (arr[i] > rightmost_element) {
 
                // Push arr[i] into the sequence
                sequence.push_back(arr[i]);
 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
 
            // If arr[j] > rightmost element
            else if (arr[j] > rightmost_element) {
 
                // Push arr[j] into the sequence
                sequence.push_back(arr[j]);
 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else
                break;
        }
 
        // If arr[i] is equal to arr[j]
        else if (arr[i] == arr[j]) {
 
            // If i and j are at the same element
            if (i == j) {
 
                // If arr[i] > rightmostelement
                if (arr[i] > rightmost_element) {
 
                    // Push arr[j] into the sequence
                    sequence.push_back(arr[i]);
 
                    // Update rightmost element
                    rightmost_element = arr[i];
                    i++;
                }
                break;
            }
            else {
                sequence.push_back(arr[i]);
 
                // Traverse array
                // from left to right
                int k = i + 1;
 
                // Stores the increasing
                // sequence from the left end
                vector max_left;
 
                // Traverse array from left to right
                while (k < j && arr[k] > arr[k - 1]) {
 
                    // Push arr[k] to max_left vector
                    max_left.push_back(arr[k]);
                    k++;
                }
 
                // Traverse the array
                // from right to left
                int l = j - 1;
 
                // Stores the increasing
                // sequence from the right end
                vector max_right;
 
                // Traverse array from right to left
                while (l > i && arr[l] > arr[l + 1]) {
 
                    // Push arr[k] to max_right vector
                    max_right.push_back(arr[l]);
                    l--;
                }
 
                // If size of max_left is greater
                // than max_right
                if (max_left.size() > max_right.size())
                    for (int element : max_left)
 
                        // Push max_left elements to
                        // the original sequence
                        sequence.push_back(element);
 
                // Otherwise
                else
                    for (int element : max_right)
 
                        // Push max_right elements to
                        // the original sequence
                        sequence.push_back(element);
                break;
            }
        }
    }
 
    // Print the sequence
    for (int element : sequence)
        printf("%d ", element);
}
 
// Driver Code
int main()
{
    int N = 4;
    int arr[] = { 1, 3, 2, 1 };
 
    // Print the longest increasing
    // sequence using boundary elements
    findMaxLengthSequence(N, arr);
}

Java

// Java program for the above approach
import java.util.*;
 
class GFG{
     
// Function to find longest strictly
// increasing sequence using boundary elements
static void findMaxLengthSequence(int N, int[] arr)
{
     
    // Maintains rightmost element
    // in the sequence
    int rightmost_element = -1;
     
    // Pointer to start of array
    int i = 0;
     
    // Pointer to end of array
    int j = N - 1;
     
    // Stores the required sequence
    Vector sequence = new Vector();
     
    // Traverse the array
    while (i <= j)
    {
     
        // If arr[i]>arr[j]
        if (arr[i] > arr[j])
        {
             
            // If arr[j] is greater than
            // rightmost element of the sequence
            if (arr[j] > rightmost_element)
            {
                 
                // Push arr[j] into the sequence
                sequence.add(arr[j]);
                 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else if (arr[i] > rightmost_element)
            {
                 
                // Push arr[i] into the sequence
                sequence.add(arr[i]);
                 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
            else
            break;
        }
     
        // If arr[i] < arr[j]
        else if (arr[i] < arr[j])
        {
         
            // If arr[i] > rightmost element
            if (arr[i] > rightmost_element)
            {
             
                // Push arr[i] into the sequence
                sequence.add(arr[i]);
                 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
         
            // If arr[j] > rightmost element
            else if (arr[j] > rightmost_element)
            {
                 
                // Push arr[j] into the sequence
                sequence.add(arr[j]);
                 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else
                break;
        }
         
        // If arr[i] is equal to arr[j]
        else if (arr[i] == arr[j])
        {
         
            // If i and j are at the same element
            if (i == j)
            {
             
                // If arr[i] > rightmostelement
                if (arr[i] > rightmost_element)
                {
                     
                    // Push arr[j] into the sequence
                    sequence.add(arr[i]);
                     
                    // Update rightmost element
                    rightmost_element = arr[i];
                    i++;
                }
                break;
            }
            else
            {
                sequence.add(arr[i]);
                 
                // Traverse array
                // from left to right
                int k = i + 1;
                 
                // Stores the increasing
                // sequence from the left end
                Vector max_left = new Vector();
                 
                // Traverse array from left to right
                while (k < j && arr[k] > arr[k - 1])
                {
                     
                    // Push arr[k] to max_left vector
                    max_left.add(arr[k]);
                    k++;
                }
                     
                // Traverse the array
                // from right to left
                int l = j - 1;
                 
                // Stores the increasing
                // sequence from the right end
                Vector max_right = new Vector();
                 
                // Traverse array from right to left
                while (l > i && arr[l] > arr[l + 1])
                {
                     
                    // Push arr[k] to max_right vector
                    max_right.add(arr[l]);
                    l--;
                }
                 
                // If size of max_left is greater
                // than max_right
                if (max_left.size() > max_right.size())
                    for(int element : max_left)
                     
                        // Push max_left elements to
                        // the original sequence
                        sequence.add(element);
                 
                // Otherwise
                else
                    for(int element : max_right)
                     
                        // Push max_right elements to
                        // the original sequence
                        sequence.add(element);
                         
                break;
            }
        }
    }
     
    // Print the sequence
    for(int element : sequence)
        System.out.print(element + " ");
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 4;
    int[] arr = { 1, 3, 2, 1 };
     
    // Print the longest increasing
    // sequence using boundary elements
    findMaxLengthSequence(N, arr);
}
}
 
// This code is contributed by divyeshrabadiya07

Python3

# Python3 program for the above approach
 
# Function to find longest strictly
# increasing sequence using boundary elements
def findMaxLengthSequence(N, arr):
     
    # Maintains rightmost element
    # in the sequence
    rightmost_element = -1
 
    # Pointer to start of array
    i = 0
 
    # Pointer to end of array
    j = N - 1
 
    # Stores the required sequence
    sequence = []
 
    # Traverse the array
    while (i <= j):
 
        # If arr[i]>arr[j]
        if (arr[i] > arr[j]):
 
            # If arr[j] is greater than
            # rightmost element of the sequence
            if (arr[j] > rightmost_element):
 
                # Push arr[j] into the sequence
                sequence.append(arr[j])
 
                # Update rightmost element
                rightmost_element = arr[j]
                j -= 1
 
            elif (arr[i] > rightmost_element):
                 
                # Push arr[i] into the sequence
                sequence.append(arr[i])
 
                # Update rightmost element
                rightmost_element = arr[i]
                i += 1
 
            else:
                break
 
        # If arr[i] < arr[j]
        elif (arr[i] < arr[j]):
 
            # If arr[i] > rightmost element
            if (arr[i] > rightmost_element):
 
                # Push arr[i] into the sequence
                sequence.append(arr[i])
 
                # Update rightmost element
                rightmost_element = arr[i]
                i += 1
                 
            # If arr[j] > rightmost element
            elif (arr[j] > rightmost_element):
                 
                # Push arr[j] into the sequence
                sequence.append(arr[j])
 
                # Update rightmost element
                rightmost_element = arr[j]
                j -= 1
 
            else:
                break
 
        # If arr[i] is equal to arr[j]
        elif (arr[i] == arr[j]):
             
            # If i and j are at the same element
            if (i == j):
                 
                # If arr[i] > rightmostelement
                if (arr[i] > rightmost_element):
 
                    # Push arr[j] into the sequence
                    sequence.append(arr[i])
 
                    # Update rightmost element
                    rightmost_element = arr[i]
                    i += 1
 
                break
 
            else:
                sequence.append(arr[i])
 
                # Traverse array
                # from left to right
                k = i + 1
 
                # Stores the increasing
                # sequence from the left end
                max_left = []
 
                # Traverse array from left to right
                while (k < j and arr[k] > arr[k - 1]):
 
                    # Push arr[k] to max_left vector
                    max_left.append(arr[k])
                    k += 1
 
                # Traverse the array
                # from right to left
                l = j - 1
 
                # Stores the increasing
                # sequence from the right end
                max_right = []
 
                # Traverse array from right to left
                while (l > i and arr[l] > arr[l + 1]):
                     
                    # Push arr[k] to max_right vector
                    max_right.append(arr[l])
                    l -= 1
                     
                # If size of max_left is greater
                # than max_right
                if (len(max_left) > len(max_right)):
                    for element in max_left:
                         
                        # Push max_left elements to
                        # the original sequence
                        sequence.append(element)
 
                # Otherwise
                else:
                    for element in max_right:
                         
                        # Push max_right elements to
                        # the original sequence
                        sequence.append(element)
                break
             
    # Print the sequence
    for element in sequence:
        print(element, end = " ")
 
# Driver Code
if __name__ == '__main__':
     
    N = 4
    arr = [ 1, 3, 2, 1 ]
     
    # Print the longest increasing
    # sequence using boundary elements
    findMaxLengthSequence(N, arr)
 
# This code is contributed by mohit kumar 29

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG
{
 
  // Function to find longest strictly
  // increasing sequence using boundary elements
  static void findMaxLengthSequence(int N, int[] arr)
  {
    // Maintains rightmost element
    // in the sequence
    int rightmost_element = -1;
 
    // Pointer to start of array
    int i = 0;
 
    // Pointer to end of array
    int j = N - 1;
 
    // Stores the required sequence
    List sequence = new List();
 
    // Traverse the array
    while (i <= j) {
 
      // If arr[i]>arr[j]
      if (arr[i] > arr[j]) {
 
        // If arr[j] is greater than
        // rightmost element of the sequence
        if (arr[j] > rightmost_element) {
 
          // Push arr[j] into the sequence
          sequence.Add(arr[j]);
 
          // Update rightmost element
          rightmost_element = arr[j];
          j--;
        }
        else if (arr[i] > rightmost_element) {
 
          // Push arr[i] into the sequence
          sequence.Add(arr[i]);
 
          // Update rightmost element
          rightmost_element = arr[i];
          i++;
        }
        else
          break;
      }
 
      // If arr[i] < arr[j]
      else if (arr[i] < arr[j]) {
 
        // If arr[i] > rightmost element
        if (arr[i] > rightmost_element) {
 
          // Push arr[i] into the sequence
          sequence.Add(arr[i]);
 
          // Update rightmost element
          rightmost_element = arr[i];
          i++;
        }
 
        // If arr[j] > rightmost element
        else if (arr[j] > rightmost_element) {
 
          // Push arr[j] into the sequence
          sequence.Add(arr[j]);
 
          // Update rightmost element
          rightmost_element = arr[j];
          j--;
        }
        else
          break;
      }
 
      // If arr[i] is equal to arr[j]
      else if (arr[i] == arr[j]) {
 
        // If i and j are at the same element
        if (i == j) {
 
          // If arr[i] > rightmostelement
          if (arr[i] > rightmost_element) {
 
            // Push arr[j] into the sequence
            sequence.Add(arr[i]);
 
            // Update rightmost element
            rightmost_element = arr[i];
            i++;
          }
          break;
        }
        else {
          sequence.Add(arr[i]);
 
          // Traverse array
          // from left to right
          int k = i + 1;
 
          // Stores the increasing
          // sequence from the left end
          List max_left = new List();
 
          // Traverse array from left to right
          while (k < j && arr[k] > arr[k - 1])
          {
 
            // Push arr[k] to max_left vector
            max_left.Add(arr[k]);
            k++;
          }
 
          // Traverse the array
          // from right to left
          int l = j - 1;
 
          // Stores the increasing
          // sequence from the right end
          List max_right = new List();
 
          // Traverse array from right to left
          while (l > i && arr[l] > arr[l + 1])
          {
 
            // Push arr[k] to max_right vector
            max_right.Add(arr[l]);
            l--;
          }
 
          // If size of max_left is greater
          // than max_right
          if (max_left.Count > max_right.Count)
            foreach(int element in max_left)
 
              // Push max_left elements to
              // the original sequence
              sequence.Add(element);
 
          // Otherwise
          else
            foreach(int element in max_right)
 
              // Push max_right elements to
              // the original sequence
              sequence.Add(element);
          break;
        }
      }
    }
 
    // Print the sequence
    foreach(int element in sequence)
      Console.Write(element + " ");
  }
 
  // Driver code
  static void Main()
  {
 
    int N = 4;
    int[] arr = { 1, 3, 2, 1 };
 
    // Print the longest increasing
    // sequence using boundary elements
    findMaxLengthSequence(N, arr);
  }
}
 
// This code is contribute by divyesh072019

Javascript

 
// Javascript program for the above approach
 
// Function to find longest strictly
// increasing sequence using boundary elements
function findMaxLengthSequence(N, arr)
{
     
    // Maintains rightmost element
    // in the sequence
    let rightmost_element = -1;
     
    // Pointer to start of array
    let i = 0;
     
    // Pointer to end of array
    let j = N - 1;
     
    // Stores the required sequence
    let sequence = [];
     
    // Traverse the array
    while (i <= j)
    {
     
        // If arr[i]>arr[j]
        if (arr[i] > arr[j])
        {
         
            // If arr[j] is greater than
            // rightmost element of the sequence
            if (arr[j] > rightmost_element)
            {
             
                // Push arr[j] into the sequence
                sequence.push(arr[j]);
                 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else if (arr[i] > rightmost_element)
            {
             
                // Push arr[i] into the sequence
                sequence.push(arr[i]);
                 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
            else
            break;
        }
     
        // If arr[i] < arr[j]
        else if (arr[i] < arr[j])
        {
         
            // If arr[i] > rightmost element
            if (arr[i] > rightmost_element)
            {
             
                // Push arr[i] into the sequence
                sequence.push(arr[i]);
                 
                // Update rightmost element
                rightmost_element = arr[i];
                i++;
            }
     
            // If arr[j] > rightmost element
            else if (arr[j] > rightmost_element)
            {
             
                // Push arr[j] into the sequence
                sequence.push(arr[j]);
                 
                // Update rightmost element
                rightmost_element = arr[j];
                j--;
            }
            else
            break;
        }
     
        // If arr[i] is equal to arr[j]
        else if (arr[i] == arr[j])
        {
         
            // If i and j are at the same element
            if (i == j)
            {
             
                // If arr[i] > rightmostelement
                if (arr[i] > rightmost_element)
                {
                 
                    // Push arr[j] into the sequence
                    sequence.push(arr[i]);
                     
                    // Update rightmost element
                    rightmost_element = arr[i];
                    i++;
                }
                break;
            }
            else
            {
                sequence.push(arr[i]);
                 
                // Traverse array
                // from left to right
                let k = i + 1;
                 
                // Stores the increasing
                // sequence from the left end
                let max_left = [];
                 
                // Traverse array from left to right
                while (k < j && arr[k] > arr[k - 1])
                {
                 
                    // Push arr[k] to max_left vector
                    max_left.push(arr[k]);
                    k++;
                }
                 
                // Traverse the array
                // from right to left
                let l = j - 1;
                 
                // Stores the increasing
                // sequence from the right end
                let max_right = [];
                 
                // Traverse array from right to left
                while (l > i && arr[l] > arr[l + 1])
                {
                 
                    // Push arr[k] to max_right vector
                    max_right.push(arr[l]);
                    l--;
                }
                 
                // If size of max_left is greater
                // than max_right
                if (max_left.length > max_right.length)
                for(let element = 0;
                    element < max_left.length;
                    element++)
                 
                    // Push max_left elements to
                    // the original sequence
                    sequence.push(max_left[element]);
                 
                // Otherwise
                else
                for(let element = 0;
                    element < max_right.length;
                    element++)
                 
                    // Push max_right elements to
                    // the original sequence
                    sequence.push(max_right[element]);
                break;
            }
        }
    }
     
    // Print the sequence
    for(let element = 0;
        element < sequence.length;
        element++)
        document.write(sequence[element] + " ");
}
 
// Driver code
let N = 4;
let arr = [ 1, 3, 2, 1 ];
 
// Print the longest increasing
// sequence using boundary elements
findMaxLengthSequence(N, arr);
 
// This code is contributed by suresh07
 

输出：

1 2 3

时间复杂度： O(N)
辅助空间： O(1)

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